Question

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{cc} 8 & -2 \\ 4 & 5 \\ 2 & -5 \end{array}\right] \\ B=\left[\begin{array}{ccc} -5 & 1 & -5 \\ 8 & 3 & -2 \end{array}\right] \end{array} $$

Short answer

Dimensions: \( A = 3\times2 \), \( B = 2\times3 \), \( AB = 3\times3 \), \( BA = 2\times2 \). Products: \( AB = \begin{bmatrix} -56 & 2 & -36 \\ 20 & 19 & -30 \\ -50 & -13 & 0 \end{bmatrix} \), \( BA = \begin{bmatrix} -46 & 40 \\ 72 & 9 \end{bmatrix} \).

Step-by-step solution

Identify Dimensions of Matrix A

Matrix \( A \) is given as \( \begin{bmatrix} 8 & -2 \ 4 & 5 \ 2 & -5 \end{bmatrix} \). To identify its dimensions, count the number of rows and columns. Matrix \( A \) has 3 rows and 2 columns, so its dimensions are 3x2.

Identify Dimensions of Matrix B

Matrix \( B \) is given as \( \begin{bmatrix} -5 & 1 & -5 \ 8 & 3 & -2 \end{bmatrix} \). This matrix has 2 rows and 3 columns, thus its dimensions are 2x3.

Determine Compatibility for Matrix Multiplication AB

To multiply matrices \( A \) and \( B \), the number of columns in \( A \) must equal the number of rows in \( B \). Since \( A \) is 3x2 and \( B \) is 2x3, matrix multiplication \( AB \) is possible, and the resulting matrix \( AB \) will have dimensions 3x3.

Determine Compatibility for Matrix Multiplication BA

For matrix multiplication \( B \) \( A \), the number of columns in \( B \) must match the number of rows in \( A \). \( B \) is 2x3 and \( A \) is 3x2, hence multiplication \( BA \) is also possible. The resulting matrix \( BA \) will have dimensions 2x2.

Perform Matrix Multiplication AB

Calculate product \( AB \) as follows: \( AB = \begin{bmatrix} 8 & -2 \ 4 & 5 \ 2 & -5 \end{bmatrix} \begin{bmatrix} -5 & 1 & -5 \ 8 & 3 & -2 \end{bmatrix} \)1. First row of \( A \) times each column of \( B \): - First element: \( (8)(-5) + (-2)(8) = -40 - 16 = -56 \) - Second element: \( (8)(1) + (-2)(3) = 8 - 6 = 2 \) - Third element: \( (8)(-5) + (-2)(-2) = -40 + 4 = -36 \)2. Second row of \( A \) times each column of \( B \): - First element: \( (4)(-5) + (5)(8) = -20 + 40 = 20 \) - Second element: \( (4)(1) + (5)(3) = 4 + 15 = 19 \) - Third element: \( (4)(-5) + (5)(-2) = -20 - 10 = -30 \)3. Third row of \( A \) times each column of \( B \): - First element: \( (2)(-5) + (-5)(8) = -10 - 40 = -50 \) - Second element: \( (2)(1) + (-5)(3) = 2 - 15 = -13 \) - Third element: \( (2)(-5) + (-5)(-2) = -10 + 10 = 0 \)Thus, \( AB = \begin{bmatrix} -56 & 2 & -36 \ 20 & 19 & -30 \ -50 & -13 & 0 \end{bmatrix} \).

Perform Matrix Multiplication BA

Calculate product \( BA \) as follows: \( BA = \begin{bmatrix} -5 & 1 & -5 \ 8 & 3 & -2 \end{bmatrix} \begin{bmatrix} 8 & -2 \ 4 & 5 \ 2 & -5 \end{bmatrix} \)1. First row of \( B \) times each column of \( A \): - First element: \( (-5)(8) + (1)(4) + (-5)(2) = -40 + 4 - 10 = -46 \) - Second element: \( (-5)(-2) + (1)(5) + (-5)(-5) = 10 + 5 + 25 = 40 \)2. Second row of \( B \) times each column of \( A \): - First element: \( (8)(8) + (3)(4) + (-2)(2) = 64 + 12 - 4 = 72 \) - Second element: \( (8)(-2) + (3)(5) + (-2)(-5) = -16 + 15 + 10 = 9 \)Thus, \( BA = \begin{bmatrix} -46 & 40 \ 72 & 9 \end{bmatrix} \).

Key concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial, as it lays the foundation for further operations. The dimensions of a matrix are determined by counting its rows and columns. This is always represented in the form of "rows by columns".
Matrix dimensions are a fundamental aspect of matrix algebra, dictating many of the rules and possibilities for operations.

• A matrix with dimensions 3x2, for example, has 3 rows and 2 columns.
• This information is vital when performing operations like matrix multiplication.

Knowing the matrix dimensions of the given matrices in an exercise ensures the correct application of rules and helps to predict the results of their mathematical interactions.

Product of Matrices

Multiplying matrices is not as straightforward as multiplying numbers. This operation involves taking rows from the first matrix and performing specific calculations with columns from the second matrix. The essential concept here is that each element in the resulting matrix is computed by multiplying corresponding elements and then taking their sum.
The product of matrices operation can drastically change the dimensions, depending on the starting dimensions.

• For matrices A and B, if you're calculating the product AB, take each row in A and calculate its interaction with each column in B.
• This process involves a lot of adding and multiplying, suitable for computational tools.

Thus, the product of matrices results in a new matrix that combines elements of the original ones, according to specific multiplication rules.

Matrix Compatibility

Compatibility in matrix multiplication is a vital concept because not all matrices can be multiplied directly. A key requirement for multiplying two matrices is that the number of columns in the first must equal the number of rows in the second. Only when this condition is met can matrices be multiplied.
This compatibility rule ensures that the subsequent operations result in mathematically valid products.

• If matrix A is 3x2 and matrix B is 2x3, they are compatible for multiplication in the form AB.
• However, if B is 2x3 and A is 3x2, BA is also possible because B’s columns match A’s rows.

Understanding this concept is essential for successfully working with matrices in mathematics, making sure the right combinations of matrices are chosen for multiplication.